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# The Quest for 700: Weekly GMAT Challenge (Answer)

Yesterday, Manhattan GMAT posted a GMAT question on our blog. Today, they have followed up with the answer:

To solve this problem quickly, you might try to come up with likely values for x that would make the mean equal the median. One sort of set for which the mean equals the median is a set with values symmetrically spaced around its mean/median. The values do not have to be evenly spaced.

Three values that would make the set symmetrical are 0, 10, and 20:

{0, 4, 8, 12, 16}

{4, 8, 10, 12, 16}

{4, 8, 12, 16, 20}

We are down to choices (D) and (E). Now, can we prove that no other values of x make the mean equal the median? After all, some non-symmetrical sets have their mean equal to their median: for instance, {1, 1, 2, 2.5, 3.5}. All you need to do is make the “residuals,” or differences, around the middle value cancel out (in the case above, the values to the left of 2 are 1 & 1, leaving a total residual of -2, while the values to the right of 2 are 2.5 and 3.5, leaving a total residual of +2).

Well, we can set up three scenarios, each with a relevant equation.

(1) If x is less than or equal to 8, then the median is equal to 8. We now set the mean equal to the median:

(40 + x)/5 = 8

40 + x = 40

x = 0

(2) If x is between 8 and 12, then the median is equal to x. Again, we set the mean equal to the median:

(40 + x)/5 = x

40 + x = 5x

40 = 4x

x = 10

(3) If x is greater than 12, then the median is equal to 12. Again, we set the mean equal to the median:

(40 + x)/5 = 12

40 + x = 60

x = 20

We have now exhausted all the possibilities for x. In fact, we did not have to actually compute the values of x in each case; rather, we could have simply realized that each equation is linear in x and so would have exactly one solution. Since there are three scenarios, there are exactly three values of x that satisfy the constraint of making the mean and the median equal. Indeed, if we had started with this approach, we might have gotten to the answer more quickly.

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